Make an object follow a complex curve?
As probably one of approaches is to use one cube for this animation with multiple Follow Path constraints added. 

In the example below I added 5 different curves, and 5 different Follow Path constraints for the cube. 



I used Offset option (thus animated it) in all constraints except for the first one (and actually for the circle, for some reason). I also animated Influence option of every constraint to make every one of them to work separately.



To bake this action to keyframes and get rid of constraints you should select that cube, hit Space and enter Bake Action (or 3D View > Object > Animation > Bake Action). In the options you should select Visual Keying and (optional) Clear Keyframes - you won't need them:

 

based on this answer